Your FREE ALEKS Flashcards 2026 – 200+ Cards

Parentheses, Exponents, Multiplication/Division (left→right), Addition/Subtraction (left→right).

10 — multiply before adding: 3×2=6, then 4+6=10.

A number that can be written as a fraction of two integers (e.g., 3/4, −2, 0.25).

A number that cannot be written as a fraction; its decimal never repeats or ends (e.g., √2, π).

A whole number and its negatives: …, −2, −1, 0, 1, 2, …

The distance of x from 0; always non-negative. |−5| = 5.

Add the values and keep the sign: −3 + (−4) = −7.

Subtract and keep the larger number's sign: −7 + 4 = −3.

Add its opposite: 5 − (−2) = 5 + 2 = 7.

Same signs → positive; different signs → negative.

Divide by 100 (move two places left): 25% = 0.25.

Multiply by 100: 0.4 = 40%.

Convert and multiply: 20% of 80 = 0.20 × 80 = 16.

(new − old) ÷ old × 100.

Divide the part by the percent (as a decimal): 15 is 25% of 15 ÷ 0.25 = 60.

A comparison of two quantities by division, written a:b, a to b, or a/b.

An equation setting two ratios equal: a/b = c/d.

Cross-multiply: a/b = c/d → a·d = b·c, then isolate the unknown.

A rate 'per one': miles ÷ hours = miles per hour.

Get a common denominator, then add the numerators.

Multiply straight across (numerators and denominators), then simplify.

Multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Divide numerator by denominator: 3/4 = 0.75.

A whole number plus a fraction, like 2½; equals the improper fraction 5/2.

The largest number that divides two values evenly; GCF(12, 18) = 6.

The smallest number both values divide into; LCM(4, 6) = 12.

A whole number greater than 1 with exactly two factors: 1 and itself.

A number as a digit between 1 and 10 times a power of 10: 4,500 = 4.5 × 10³.

The 'flip' of a number: the reciprocal of 3/4 is 4/3; n × (1/n) = 1.

Look at the next digit: 5 or more rounds up, less than 5 rounds down.

Round numbers to sanity-check an answer — vital since ALEKS has no answer choices.

22 — exponent first (16), then ÷2 = 8, then 24 − 8 + 6 = 22.

3/5 (0.6 = 6/10 = 3/5).

37.5% (3 ÷ 8 = 0.375).

3/4 — divide top and bottom by the GCF, 6.

The power of 10 that best describes a number's size.

A percent added to a cost to set a price; new price = cost × (1 + rate).

A percent subtracted from a price; sale price = price × (1 − rate).

I = P · r · t (principal × rate × time).

Add the values and divide by how many there are.

60 inches — multiply by 12 (a proportion: 12 in / 1 ft).

A fraction whose numerator ≥ denominator, like 7/4.

The number that adds to give 0: the opposite of 7 is −7.

An equation with variables only to the first power; graph is a straight line (2x + 3 = 11).

Isolate the variable with inverse operations, keeping both sides balanced.

Whatever you do to one side, do to the other.

a(b + c) = ab + ac — distribute before combining like terms.

Add terms with the same variable and power: 3x + 5x = 8x.

Multiply every term by the common denominator.

Compares expressions with <, >, ≤, or ≥; its solution is a range of values.

You multiply or divide both sides by a negative number.

x > −3 — dividing by −2 flips < to >.

Open = strict (< or >); closed = inclusive (≤ or ≥).

Two inequalities joined by 'and' (overlap) or 'or' (union).

Gives two cases: x = a or x = −a (for a ≥ 0).

2x − 5 = 9 or 2x − 5 = −9 → x = 7 or x = −2.

Two or more equations solved together; the solution satisfies all of them.

Solve one equation for a variable and plug it into the other.

Add or subtract the equations to cancel a variable.

The point where the two lines intersect.

An equation with a squared variable: a x² + b x + c = 0.

x = (−b ± √(b² − 4ac)) ÷ 2a — solves any quadratic.

Positive → 2 real roots; zero → 1; negative → none (real).

If A·B = 0, then A = 0 or B = 0 — the basis for solving by factoring.

Set = 0, factor, then set each factor to 0.

If x² = k, then x = ±√k (use when there's no bx term).

An equation with several variables; solve for one in terms of the others.

Substitute it back into the ORIGINAL equation.

x = x+1 (false) → no solution; 0 = 0 (true) → infinitely many.

x = 6 — distribute: 5x − 11 = 2x + 7 → 3x = 18.

x ≤ 4 — add 1, divide by 2 (positive, no flip).

Open circle at 3, shaded to the right.

The number multiplying a variable: in 7x, the coefficient is 7.

A term with no variable, like the 5 in 3x + 5.

(3, 2) — add the equations to eliminate y.

Lines are parallel (no intersection) → no solution.

Lines are the same → infinitely many solutions.

Rewrite a quadratic as (x − h)² = k to solve or find the vertex.

x = ±7 (square-root method).

Rise over run: (y₂ − y₁) ÷ (x₂ − x₁); the m in y = mx + b.

A rule giving exactly one output for each input; written f(x).

A graph is a function if no vertical line crosses it more than once.

The set of allowed inputs (x-values) of a function.

The set of resulting outputs (y-values) of a function.

Substitute the input for x: if f(x) = 2x + 1, then f(3) = 7.

Where a graph crosses the y-axis (x = 0); the b in y = mx + b.

Where a graph crosses the x-axis (y = 0); also called a zero or root.

y = mx + b — read the slope m and intercept b directly.

0 (no rise).

Undefined (no run).

Positive rises left→right; negative falls left→right.

Same slope, different intercepts.

Slopes are negative reciprocals (2 and −½).

y − y₁ = m(x − x₁) — build a line from a point and a slope.

f(x) = a x² + b x + c; its graph is a parabola.

The U-shaped graph of a quadratic; has a vertex and an axis of symmetry.

Its turning point; x-coordinate = −b ÷ (2a).

The vertical line x = −b ÷ (2a) that splits a parabola in half.

Up if a > 0 (minimum); down if a < 0 (maximum).

The x-values where f(x) = 0 (the x-intercepts).

A function whose graph is a straight line: f(x) = mx + b.

Outputs rise as inputs rise (graph goes up left→right).

Outputs fall as inputs rise (graph goes down left→right).

2 — (8 − 2)/(4 − 1) = 6/3.

Ax + By = C.

Max if it opens down (a < 0); min if it opens up (a > 0).

Names the output for input x; f(2) is the output when x = 2.

No — the input 1 maps to two different outputs.

Shows repeated multiplication: 2³ = 2 × 2 × 2 = 8.

xᵃ × xᵇ = xᵃ⁺ᵇ — multiply like bases by adding exponents.

xᵃ ÷ xᵇ = xᵃ⁻ᵇ — divide like bases by subtracting exponents.

(xᵃ)ᵇ = xᵃᵇ — a power to a power multiplies exponents.

x⁰ = 1 for any nonzero x.

x⁻ⁿ = 1 ÷ xⁿ — take the reciprocal.

(xy)ⁿ = xⁿyⁿ.

x⁸ — add the exponents.

A sum of terms with whole-number powers, like 3x² − 5x + 2.

The highest exponent on the variable.

1, 2, or 3 terms, respectively.

Combine like terms (same variable and power).

Distribute the minus sign, then combine like terms.

Multiply two binomials: First, Outer, Inner, Last.

x² + 7x + 12 (by FOIL).

Rewriting an expression as a product of factors — the reverse of multiplying.

Always pull out the greatest common factor before other factoring.

Find two numbers that multiply to c and add to b.

(x + 4)(x + 5) — 4·5 = 20, 4 + 5 = 9.

a² − b² = (a + b)(a − b).

a² + 2ab + b² = (a + b)².

Set = 0, factor, apply the zero-product property.

The number in front of the highest-degree term.

Terms with identical variables and exponents; only these combine.

8 — 25 − 20 + 3.

8x³ — cube both the 2 and the x.

x⁸ — multiply the exponents.

1/x².

x² + 5x (distribute).

3x(2x + 3) — pull out the GCF 3x.

(x + 4)(x − 4) — difference of squares.

A polynomial with three terms.

Terms written in order of decreasing degree.

A fraction whose numerator and denominator are polynomials.

Any value that makes the denominator zero — it's excluded (undefined).

Factor top and bottom, then cancel common factors.

Factor, cancel, then multiply straight across.

Multiply by the reciprocal of the second fraction.

Get a common denominator first, then add numerators.

The least common multiple of the denominators.

Multiply through by the LCD to clear fractions, then solve.

An answer that fails the original equation (often makes a denominator 0).

To discard extraneous answers that make a denominator zero.

A fraction with fractions inside; simplify by multiplying by a common denominator.

A function f(x) = polynomial ÷ polynomial; may have vertical asymptotes.

A line x = a where the denominator is 0 (and the numerator isn't).

All real numbers except where the denominator equals 0.

x = 2 (it makes the denominator zero).

x + 3 — factor the top to (x+3)(x−3) and cancel (x ≠ 3).

ad = bc.

A proportion is two equal ratios; both clear by cross-multiplying.

An expression with a root, like √12 (a fractional exponent means a root).

A number that, multiplied by itself, gives the value: √25 = 5.

2√3 — factor out the perfect square: √(4·3) = 2√3.

Combine only LIKE radicals (same root of the same number).

√a × √b = √(ab).

√a ÷ √b = √(a/b).

A fractional power IS a root: $x^{1/n}$ equals the nth root of x, and $x^{1/2}=\sqrt{x}$.

The nth root of x, raised to the m power: $\left(\sqrt[n]{x}\right)^m$.

Multiply top and bottom to remove the root from the denominator.

The number that cubed gives the value: ∛27 = 3.

ⁿ√x reverses raising to the nth power.

An equation with a variable under a root; isolate the radical and square both sides.

Squaring can create extraneous solutions — verify each in the original.

|a| — the principal (non-negative) square root.

5√2 — √(25·2).

a (for a ≥ 0).

The small number on the root; ∛ has index 3 (cube root).

2 — it is $\sqrt[3]{8}=2$.

x = 25 — square both sides, then check.

y = a·bˣ — the variable is in the exponent; models growth/decay.

Growth when the base b > 1; decay when 0 < b < 1.

The inverse of an exponential: log_b(x) = y means bʸ = x.

3, because 2³ = 8.

Logarithm with base 10, written log.

Logarithm with base e (e ≈ 2.718).

log(MN) = log M + log N.

log(M/N) = log M − log N.

log(Mᵖ) = p · log M.

log_b(b) = 1 and log_b(1) = 0.

Rewrite in exponential form, or condense to one log and match the insides.

x = 13 — rewrite as 2⁴ = x + 3, so x + 3 = 16.

The argument must be positive (> 0).

Use g's output as f's input — work inside-out.

Undoes a function; swap x and y and solve for y.

Reflects the original across the line y = x.

One-to-one functions (those passing the horizontal line test).

They are inverse functions — each undoes the other.

log_b(x) = log(x) ÷ log(b).

A function is one-to-one (invertible) if no horizontal line hits it twice.

log₅ x − log₅ 7 (quotient rule).

log₃ 20 (product rule).

6 — f(7)=3, then g(3)=6.

f⁻¹(x) = 2(x − 5).

An irrational constant ≈ 2.718, the base of the natural log.

The horizontal line the curve approaches but never touches (often y = 0).

The total distance around a 2-D figure.

The surface a 2-D figure covers, in square units.

The space a 3-D solid fills, in cubic units.

length × width.

½ × base × height.

πr².

2πr (or πd).

length × width × height.

πr²h.

(4/3)πr³.

The radius is half the diameter (d = 2r).

a² + b² = c², where c is the hypotenuse of a right triangle.

The longest side of a right triangle, opposite the right angle.

√[(x₂ − x₁)² + (y₂ − y₁)²].

((x₁ + x₂)/2, (y₁ + y₂)/2).

180°.

360°.

Two angles that add to 90°.

Two angles that add to 180°.

Opposite angles formed by two intersecting lines; they are equal.

Same shape, proportional sides, equal angles.

Same shape AND same size.

sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

Opposite side ÷ hypotenuse.

Adjacent side ÷ hypotenuse.

Opposite side ÷ adjacent side.

sin²θ + cos²θ = 1.

sin⁻¹, cos⁻¹, tan⁻¹ find an angle from a ratio.

180° = π radians; multiply degrees by π/180 to convert.

Exactly 90°, marked with a small square.

16π square units.

14π units.

42 square units (½·14·6).

Less than 90° / exactly 90° / between 90° and 180°.

Two equal sides and two equal base angles.

All three sides equal; all angles 60°.

No equal sides.

base × height.

½ × (base₁ + base₂) × height.

The total area of all faces/surfaces of a 3-D solid.

A chord through the center; twice the radius.

Sides in ratio 1 : √3 : 2.

Sides in ratio 1 : 1 : √2.

I (+,+), II (−,+), III (−,−), IV (+,−), counterclockwise.